The development of quantum measurement theory, initiated by von Neumann, only indicated a possibility for resolution of the interpretational crisis of quantum mechanics. We do this by divorcing the algebra of the dynamical generators and the algebra of the actual observables, or beables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the past beables with the potential future. This rule, together with the self-compatibility of the measurements insuring the consistency of the histories, is called the nondemolition, or causality principle in modern quantum theory. The application of this rule in the form of the dynamical commutation relations leads in particular to the derivation of the von Neumann projection postulate. This gives a quantum stochastic solution, in the form of the dynamical filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schroedinger and Bohr.
The development of quantum measurement theory, initiated by von Neumann, only indicated a possibility for resolution of the interpretational crisis of quantum mechanics. We do this by divorcing the algebra of the dynamical generators and the algebra of the actual observables, or beables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the past beables with the potential future. This rule, together with the self-compatibility of the measurements insuring the consistency of the histories, is called the nondemolition, or causality principle in modern quantum theory. The application of this rule in the form of the dynamical commutation relations leads in particular to the derivation of the von Neumann projection postulate. This gives a quantum stochastic solution, in the form of the dynamical filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schroedinger and Bohr.
A history of the discovery of quantum mechanics and paradoxes of its interpretation is reconsidered from the modern point of view of quantum stochastics and information. It is argued that in the orthodox quantum mechanics there is no place for quantum phenomenology such as events. The development of quantum measurement theory, initiated by von Neumann, and Bell's conceptual critics of hidden variable theories indicated a possibility for resolution of this crisis. This can be done by divorcing the algebra of the dynamical generators and an extended algebra of the potential (quantum) and the actual (classical) observables. The latter, called beables, form the center of the algebra of all observables, as the only visible (macroscopic) observables must be compatible with any hidden (microscopic) observable. It is shown that within this approach quantum causality can be rehabilitated within an extended quantum mechanics (eventum mechanics) in the form of a superselection rule for compatibility of the consistent histories with the statistically predictable future. The application of this rule in the form of the nondemolition principle leads to the statistical inference of the von Neumann projection postulate, and also to the more general quantum information dynamics for instantaneous events, spontaneous localizations (i.e. quantum jumps), and state diffusions (i.e. continuous trajectories). This gives a dynamical solution, in the form of a Dirac boundary value problem and reduced filtering equations, of the notorious decoherence and measurement problems which was tackled unsuccessfully by many famous mathematicians and physicists starting with von Neumann, Schroedinger and Bohr.
It is shown that many dissipative phenomena of "old" quantum mechanics which appeared 100 years ago in the form of the statistics of quantum thermal noise and quantum spontaneous jumps, have never been explained by the "new" conservative quantum mechanics discovered 75 years ago by Heisenberg and Schroedinger. This led to numerous quantum paradoxes which are reconsidered in this paper. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for resolution of this interpretational crisis by divorcing the algebra of the dynamical generators from the algebra of the actual observables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of past observables with the potential future. This rule, together with the self-compatibility of measurements insuring the consistency of histories, is called the nondemolition principle. The application of this causality condition in the form of the dynamical commutation relations leads to the derivation of the generalized von Neumann reductions, usharp, instantaneous, spontaneous, and even continuous in time. This gives a quantum probabilistic solution, in the form of the dynamical filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schroedinger and Bohr. The simplest Markovian quantum stochastic model for the continuous in time measurements involves a boundary-value problem in second quantization for input "offer" waves in one extra dimension, and a reduction of the algebra of "actual" observables to an Abelian subalgebra for the output waves.
A history of the discovery of `new' quantum mechanics and the paradoxes of its probabilistic interpretation are briefly reviewed from the modern point of view of quantum probability and information. Modern quantum theory, which has been developed during the last 20 years for the treatment of quantum open systems including quantum noise, decoherence, quantum diffusions and spontaneous jumps occurring under continuous in time observation, is not yet a part of the standard curriculum of quantum physics. It is argued that the conventional formalism of quantum mechanics is insufficient for the description of quantum events, such as spontaneous decays say, and the new experimental phenomena related to individual quantum measurements, but they have all received an adequate mathematical treatment in quantum stochastics of open systems. Moreover, the only reasonable probabilistic interpretation of quantum mechanics put forward by Max Born was, in fact, in irreconcilable contradiction with traditional mechanical reality and causality. This led to numerous quantum paradoxes, some of them due to the great inventors of quantum theory such as Einstein and Schrödinger. They are reconsidered in this paper from the modern point of view of quantum stochastics and information. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for resolution of this interpretational crisis by divorcing the algebra of the dynamical generators and the algebra of the actual observables, or Bell's be ables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the actual histories with the potential future. This rule, together with the self-compatibility of the measurements ensuring the consistency of the histories, is called the nondemolition, or causality principle in modern quantum theory. The application of this rule in the form of dynamical commutation relations leads to the derivation of the von Neumann projection postulate, and also to the more general reductions, instantaneous, spontaneous, and even continuous in time. This gives a dynamical solution, in the form of the quantum stochastic filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schrödinger and Bohr. It has been recently proved that the quantum stochastic model for the continuous in time measurements is equivalent to a Dirac type boundary-value problem for the secondary quantized input `offer waves from future' in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the `trajectories of the output particles'. This supports the corpuscular-wave duality in the form of the thesis that everything in the future are quantized waves, while everything in the past are trajectories of the recorded particles.
Abstract. We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous– spectrum and continuous-in-time observations. The “collapsed state–vector” after the “objectification ” is simply treated as a random vector of the a posteriori state given by the quantum filtering, i.e., the conditioning of the a priori induced state on the corresponding reduced algebra. The nonlinear phenomenological equation of “continuous spontaneous localization ” has been derived from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments. 1. The status of quantum measurement theory
In this article, we introduce a new form of quantum selective measurement in which the von Neumann projection postulate is replaced by quasilinear evolution, governed by a nonlinear generalization of the von Neumann equation. We demonstrate that this equation preserves the equivalence of quantum ensembles and, consequently, satisfies the no-signalling principle, ensuring consistency with both quantum mechanics and Einstein causality. Our approach eliminates the need for instantaneous, discontinuous state collapse and provides a unified description of the postmeasurement quantum state reduction as a form of quantum state evolution. Notably, it does not require invoking concepts such as the quantum state assigned to a classical apparatus. At the same time, the stochastic character of selective measurement and the Born rule remain unchanged. We present several numerical solutions of the evolution equation for quasilinear selective measurement in two-level quantum systems and compare them with the standard von Neumann projection. The results demonstrate agreement between the two measurement schemes in their fundamental properties. Furthermore, we investigate phenomena associated with the structural instability of the evolution equation and identify very narrow parameter regions in which the outcomes deviate from those predicted by the von Neumann projection. These regions may offer opportunities to test the proposed approach experimentally. Finally, using specific analytical solutions, we discuss the Stern-Gerlach experiment within the framework of quasilinear measurement.
The quantum measurement problem, i.e., the apparent conflict between unitary quantum evolution and non-unitary, stochastic wave-function collapse, remains unresolved a century after the formulation of quantum mechanics. We first review the standard picture, from the Copenhagen prescription with its Heisenberg cut, through von Neumann's movable cut, to environment-induced decoherence, which explains the emergence of stable classical records but still presupposes the Born rule and therefore cannot by itself replace the measurement postulates. We then connect this problem to the Maximum Entropy Principle in two complementary ways. First, decoherence drives the measurement apparatus toward the least biased state compatible with the dynamically protected pointer distribution, so that the emergent collapse may be viewed as a thermodynamic relaxation toward constrained maximum entropy. Second, we propose a Timeless Histories formulation of quantum mechanics in which the primitive objects are ordered sequences of events rather than evolving wave functions. Their conditional probabilities are assigned, relative to specified refinements, by a single Born-Boltzmann rule that combines the quadratic structure of Born's rule with Boltzmann's counting of equally probable microstates. The textbook form of Born's rule, the projection postulate, density operators, and wave functions are then recovered as derived informational constructs, while collapse becomes Bayesian conditioning on recorded events. Time and space enter only subsequently through unitary translation symmetry. Since the probability rule is postulated without reference to any physical collapse or classical observer, the usual circularity of the decoherence program is avoided. Decoherence can then be invoked to explain the stability and approximate additivity of macroscopic records, i.e., the emergence of classicality.